[Paper Review] Decentralised adaptive-gain control for eliminating epidemic spreading on networks

Liam Walsh, Mengbin Ye+2 more|arXiv (Cornell University)|May 26, 2023

Complex Network Analysis Techniques4 citations

TL;DR

This paper proposes decentralized adaptive-gain control laws for eliminating epidemic spreading in networked SIS models, where each node independently adjusts its control gain via a differential equation to reduce infection or increase recovery rates. The method ensures global disease elimination with finite, positive gains, even when only a subset of nodes are controlled, provided a necessary and sufficient condition on network topology is satisfied.

ABSTRACT

This paper considers the classical Susceptible--Infected--Susceptible (SIS) network epidemic model, which describes a disease spreading through $n$ nodes, with the network links governing the possible transmission pathways of the disease between nodes. We consider feedback control to eliminate the disease in scenarios where the disease would otherwise persist in an uncontrolled network. We propose a family of decentralised adaptive-gain control algorithms, in which each node has a control gain that adaptively evolves according to a differential equation, independent of the gains of other nodes. The adaptive gain is applied multiplicatively to either decrease the infection rate or increase the recovery rate. To begin, we assume all nodes are controlled, and prove that both infection rate control and recovery rate control algorithms eliminate the disease with the limiting gains being positive and finite. Then, we consider the possibility of controlling a subset of the nodes, for both the infection rate control and recovery rate control. We first identify a necessary and sufficient condition for the existence of a subset of nodes, which if controlled would result in the elimination of the disease. For a given network, there may exist several such viable subsets, and we propose an iterative algorithm to identify such a subset. Simulations are provided to demonstrate the effectiveness of the various proposed controllers.

Motivation & Objective

• To address the limitations of static, centralized control in epidemic networks by proposing dynamic, decentralized feedback control.
• To enable disease elimination in SIS network models where the basic reproduction number R₀ > 1, using only local information at each node.
• To identify minimal control subsets that can eradicate the disease, even when full-network control is not feasible.
• To design adaptive gain mechanisms that evolve continuously and independently per node, ensuring robustness and scalability.
• To provide a systematic algorithm for identifying viable control node subsets based on network structure and cycle gains.

Proposed method

• Each node employs a decentralized adaptive gain governed by a differential equation that evolves independently of other nodes’ gains.
• The control gain multiplicatively modifies either the infection rate (infection rate control) or the recovery rate (recovery rate control) at each node.
• For full-network control, the adaptive gain dynamics ensure convergence to a positive, finite limit, driving the system to the disease-free equilibrium.
• For partial control, a two-stage algorithm identifies a minimal set of nodes to control: first, nodes with dᵢ < bᵢᵢ are required to be controlled; second, strongly connected components with cycle gains ≥1 are analyzed to identify critical nodes.
• The spectral abscissa of the matrix −D₁ + B₁₁ is used to assess stability of the uncontrolled subsystem, ensuring Hurwitz properties after node selection.
• The method leverages graph-theoretic analysis of simple cycles and sum-cycle gains to determine which nodes must be included in the control set.

Experimental results

Research questions

• RQ1 Can decentralized adaptive-gain control eliminate the disease in a networked SIS model without requiring global network knowledge?
• RQ2 What is the minimal subset of nodes whose control can eradicate the disease, and how can such a subset be systematically identified?
• RQ3 How do network topology and cycle structure influence the convergence and effectiveness of decentralized adaptive control?
• RQ4 What conditions ensure that adaptive gains remain positive and finite under decentralized control?
• RQ5 How does the choice of controlled nodes affect the convergence rate of disease elimination?

Key findings

• In full-network control, both infection rate and recovery rate control algorithms drive the system to the disease-free equilibrium with adaptive gains converging to strictly positive and finite values.
• For partial control, the proposed two-stage algorithm successfully identifies a minimal set of nodes whose control ensures disease elimination, based on node degree and cycle gain analysis.
• In simulations, controlling nodes a and d or a and f resulted in complete disease elimination with fast convergence, while controlling only node a failed to eliminate the disease from the entire network.
• The convergence rate varied significantly depending on the choice of controlled nodes, with up to an order of magnitude difference observed, highlighting the role of network structure in control performance.
• When only node a was controlled, the gain gₐ(t) converged to zero, and the disease remained endemic in nodes c, d, e, f, confirming the necessity of additional controlled nodes.
• The spectral abscissa of −D₁ + B₁₁ dropped below zero after proper node selection, confirming the stability of the uncontrolled subsystem and the feasibility of disease eradication.